Selected and recent papers:

To understand deep learning we need to understand kernel learning
[arxiv]
Mikhail Belkin, Siyuan Ma, Soumik Mandal.

+ abstract

Generalization performance of classifiers in deep learning has recently become a subject of intense study. Deep models, which are typically heavily overparametrized, tend to fit the training data exactly. Despite this overfitting, they perform well on test data, a phenomenon not yet fully understood.
The first point of our paper is that strong performance of overfitted classifiers is not a unique feature of deep learning. Using six realworld and two synthetic datasets, we establish experimentally that kernel classifiers trained to have zero classification error (overfitting) or zero regression error (interpolation) perform very well on test data.
We proceed to prove lower bounds on the norm of overfitted solutions for smooth kernels, showing that they increase nearly exponentially with data size. Since most generalization bounds depend polynomially on the norm of the solution, this result implies that they diverge as data increases. Furthermore, the existing bounds do not apply to interpolated classifiers.
We also show experimentally that (nonsmooth) Laplacian kernels easily fit random labels using a version of SGD,
a finding that parallels results recently reported for ReLU neural networks. In contrast, as expected from theory, fitting noisy data requires many more epochs for smooth Gaussian kernels. The observation that the ultimate performance of overfitted Laplacian and Gaussian classifiers on the test is quite similar, suggests that generalization is tied to the properties of the kernel function rather than the optimization process.
We see that some key phenomena of deep learning are manifested similarly in kernel methods in the ``modern'' overfitted regime. We argue that progress on understanding deep learning will be difficult, until more analytically tractable ``shallow'' kernel methods are better understood.
The combination of the experimental and theoretical results presented in this paper indicates a need for new theoretical ideas for understanding properties of classical kernel methods.

The power of interpolation: understanding the effectiveness of SGD in modern overparametrized learning
[arxiv]
Siyuan Ma, Raef Bassily, Mikhail Belkin.

+ abstract

Stochastic Gradient Descent (SGD) with small minibatch is a key
component in modern largescale machine learning. However, its
efficiency has not been easy to analyze as most theoretical results
require adaptive rates and show convergence rates far slower than that
for gradient descent, making computational comparisons difficult.
In this paper we aim to clarify the issue of fast SGD convergence. The
key observation is that most modern architectures are
overparametrized and are trained to interpolate the data by driving
the empirical loss (classification and regression) close to zero.
While it is still unclear why these interpolated solutions perform
well on test data, these regimes allow for very fast convergence of
SGD, comparable in the number of iterations to gradient descent.
Specifically, consider the setting with quadratic objective function,
or near a minimum, where the quadratic term is dominant. We show that:
(1) Minibatch size 1with constant step size is optimal in terms of
computations to achieve a given error. (2) There is a critical
minibatch size such that: (a:linear scaling) SGD iteration with
minibatch size m smaller than the critical size is nearly equivalent
to m iterations of minibatch size 1. (b:saturation) SGD iteration
with minibatch larger than the critical size is nearly equivalent to
a gradient descent step.
The critical minibatch size can be viewed as
the limit for effective minibatch parallelization. It is also nearly
independent of the data size, implying O(n) acceleration over GD per
unit of computation.
We give experimental evidence on real data, with the results closely
following our theoretical analyses.
Finally, we show how the interpolation perspective and our results fit
with recent developments in training deep neural networks and discuss
connections to adaptive rates for SGD and variance reduction.

Approximation beats concentration? An approximation view on inference with smooth kernels [arxiv]
Mikhail Belkin

+ abstract

Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data.
In this paper we take the approximation theory point of view to explore various aspects of smooth kernels related to their inferential properties. We analyze eigenvalue decay of kernels operators and matrices, properties of eigenfunctions/eigenvectors and "Fourier" coefficients of functions in the kernel space restricted to a discrete set of data points. We also investigate the fitting capacity of kernels, giving explicit bounds on the fat shattering dimension of the balls in Reproducing Kernel Hilbert spaces. Interestingly, the same properties that make kernels very effective approximators for functions in their "native" kernel space, also limit their capacity to represent arbitrary functions. We discuss various implications, including those for gradient descent type methods.
It is important to note that most of our bounds are measure independent. Moreover, at least in moderate dimension, the bounds for eigenvalues are much tighter than the bounds which can be obtained from the usual matrix concentration results. For example, we see that the eigenvalues of kernel matrices show nearly exponential decay with constants depending only on the kernel and the domain. We call this "approximation beats concentration" phenomenon as even when the data are sampled from a probability distribution, some of their aspects are better understood in terms of approximation theory.

Diving into the shallows: a computational perspective on largescale shallow learning [arxiv, EigenPro code (Keras/Matlab)]
Siyuan Ma, Mikhail Belkin, NIPS 2017 (spotlight, 5% of submissions).

+ abstract
 In this paper we first identify a basic limitation in gradient descentbased optimization methods when used in conjunctions with smooth kernels. An analysis based on the spectral properties of the kernel demonstrates that only a vanishingly small portion of the function space is reachable after a polynomial number of gradient descent iterations. This lack of approximating power drastically limits gradient descent for a fixed computational budget leading to serious overregularization/underfitting. The issue is purely algorithmic, persisting even in the limit of infinite data.
To address this shortcoming in practice, we introduce EigenPro iteration, based on a preconditioning scheme using a small number of approximately computed eigenvectors. It can also be viewed as learning a new kernel optimized for gradient descent. It turns out that injecting this small (computationally inexpensive and SGDcompatible) amount of approximate secondorder information leads to major improvements in convergence. For large data, this translates into significant performance boost over the standard kernel methods. In particular, we are able to consistently match or improve the stateoftheart results recently reported in the literature with a small fraction of their computational budget.
Finally, we feel that these results show a need for a broader computational perspective on modern largescale learning to complement more traditional statistical and convergence analyses. In particular, many phenomena of largescale highdimensional inference are best understood in terms of optimization on infinite dimensional Hilbert spaces, where standard algorithms can sometimes have properties at odds with finitedimensional intuition. A systematic analysis concentrating on the approximation power of such algorithms within a budget of computation may lead to progress both in theory and practice.

Unperturbed: spectral analysis beyond DavisKahan [arxiv]
Justin Eldridge, Mikhail Belkin, Yusu Wang, ALT 2018

+ abstract
 Classical matrix perturbation results, such as Weyl's theorem for eigenvalues and the DavisKahan theorem for eigenvectors, are general purpose. These classical bounds are tight in the worst case, but in many settings suboptimal in the typical case. In this paper, we present perturbation bounds which consider the nature of the perturbation and its interaction with the unperturbed structure in order to obtain significant improvements over the classical theory in many scenarios, such as when the perturbation is random. We demonstrate the utility of these new results by analyzing perturbations in the stochastic blockmodel where we derive much tighter bounds than provided by the classical theory. We use our new perturbation theory to show that a very simple and natural clustering algorithm  whose analysis was difficult using the classical tools  nevertheless recovers the communities of the blockmodel exactly even in very sparse graphs.

Graphons, mergeons, and so on! [arxiv, 3min preNIPS video, NIPS video]
Justin Eldridge, Mikhail Belkin, Yusu Wang, NIPS 2016 (oral presentation, 2% of submissions)

+ abstract

In this work we develop a theory of hierarchical clustering for graphs. Our modeling assumption is that graphs are sampled from a graphon, which is a powerful and general model for generating graphs and analyzing large networks. Graphons are a far richer class of graph models than stochastic blockmodels, the primary setting for recent progress in the statistical theory of graph clustering. We define what it means for an algorithm to produce the "correct" clustering, give sufficient conditions in which a method is statistically consistent, and provide an explicit algorithm satisfying these properties.

Clustering with Bregman Divergences: an Asymptotic Analysis [link]
Chaoyue Liu, Mikhail Belkin, NIPS 2016

+ abstract

Clustering, in particular kmeans clustering, is a central topic in data analysis. Clustering with Bregman divergences is a recently proposed generalization of kmeans clustering which has already been widely used in applications. In this paper we analyze theoretical properties of Bregman clustering when the number of the clusters k is large. We establish quantization rates and describe the limiting distribution of the centers as k tends to infinity, extending wellknown results for kmeans clustering.

Eigenvectors of Orthogonally Decomposable Functions [arxiv]
Mikhail Belkin, Luis Rademacher, James Voss, Siam Journal on Computing (SICOMP) 2018, to appear, short version COLT 2016 (Learning a Hidden Basis Through Imperfect Measurements: An Algorithmic Primitive)

+ abstract

n this paper, we generalize the eigendecomposition of quadratic forms (symmetric matrices) to a broad class of "orthogonally decomposable" functions. We focus on extending two characterizations of eigenvectors: First, that the eigenvectors of a quadratic form arise from the optima structure of the quadratic form on the sphere, and second that the eigenvectors are the fixed points of the matrix power iteration. We identify a key role of convexity in extending these characterizations to our setting. The generalized power iteration is a simple first order method which we call gradient iteration. Further, our framework captures as special cases recent methods for inferential problems in machine learning in areas including orthogonal tensor decompositions, Independent Component Analysis (ICA), topic modeling, spectral clustering, and Gaussian mixture learning.
We provide a complete theoretical analysis of gradient iteration using the structure theory of discrete dynamical systems to show almost sure convergence and fast (superlinear) convergence rates. The analysis extends to the case when the observed function is only approximately orthogonally decomposable, with bounds that are polynomial in dimension and other relevant parameters, such as perturbation size. Our perturbation results can be considered as a nonlinear version of the classical DavisKahan theorem for perturbations of eigenvectors of symmetric matrices.

Back to the future: Radial Basis Function networks revisited [link]
Qichao Que, Mikhail Belkin, AI & Statistics 2016.

+ abstract

Radial Basis Function (RBF) networks are a classical family of algorithms for supervised learning. The most popular approach for training RBF networks has relied on kernel methods using regularization based on a norm in a Reproducing Kernel Hilbert Space (RKHS), which is a principled and empirically successful framework. In this paper we aim to revisit some of the older approaches to training the RBF networks from a more modern perspective. Specifically, we analyze two common regularization procedures, one based on the square norm of the coefficients in the network and another on using centers obtained by kmeans clustering. We show that both of these RBF methods can be recast as certain datadependent kernels. We provide a theoretical analysis of these methods as well as a number of experimental results, pointing out very competitive experimental performance as well as certain advantages over the standard kernel methods in terms of both flexibility (incorporating of unlabeled data) and computational complexity. Finally, our results shed light on some impressive recent successes of using soft kmeans features for image recognition and other tasks.

The Hidden Convexity of Spectral Clustering [arxiv]
James Voss, Mikhail Belkin, Luis Rademacher, AAAI 2016 (oral presentation)

+ abstract

In recent years, spectral clustering has become a standard method for data analysis used in a broad range of applications. In this paper we propose a new class of algorithms for multiway spectral clustering based on optimization of a certain "contrast function" over a sphere. These algorithms are simple to implement, efficient and, unlike most of the existing algorithms for multiclass spectral clustering, are not initializationdependent. Moreover, they are applicable without modification for normalized and unnormalized clustering, which are two common variants of spectral clustering.
Geometrically, the proposed algorithms can be interpreted as recovering a discrete weighted simplex by means of function optimization. We give complete necessary and sufficient conditions on contrast functions for the optimization to guarantee recovery of clusters. We show how these conditions can be interpreted in terms of certain "hidden convexity" of optimization over a sphere.

Learning Privately from Multiparty Data [arxiv]
Jihun Hamm, Paul Cao, Mikhail Belkin, ICML 2016

+ abstract

Learning a classifier from private data collected by multiple parties is an important problem that has many potential applications. How can we build an accurate and differentially private global classifier by combining locallytrained classifiers from different parties, without access to any party's private data? We propose to transfer the `knowledge' of the local classifier ensemble by first creating labeled data from auxiliary unlabeled data, and then train a global εdifferentially private classifier. We show that majority voting is too sensitive and therefore propose a new risk weighted by class probabilities estimated from the ensemble. Relative to a nonprivate solution, our private solution has a generalization error bounded by O(ε^2M^2) where M is the number of parties. This allows strong privacy without performance loss when M is large, such as in crowdsensing applications. We demonstrate the performance of our method with realistic tasks of activity recognition, network intrusion detection, and malicious URL detection.

A PseudoEuclidean Iteration for Optimal Recovery in Noisy ICA [link]
James Voss, Mikhail Belkin, Luis Rademacher, NIPS 2015

+abstract

Independent Component Analysis (ICA) is a popular model for blind signal separation. The ICA model assumes that a number of independent source signals are linearly mixed to form the observed signals. We propose a new algorithm, PEGI (for pseudoEuclidean Gradient Iteration), for provable model recovery for ICA with Gaussian noise. The main technical innovation of the algorithm is to use a fixed point iteration in a pseudoEuclidean (indefinite "inner product") space. The use of this indefinite "inner product" resolves technical issues common to several existing algorithms for noisy ICA. This leads to an algorithm which is conceptually simple, efficient and accurate in testing.
Our second contribution is combining PEGI with the analysis of objectives for optimal recovery in the noisy ICA model. It has been observed that the direct approach of demixing with the inverse of the mixing matrix is suboptimal for signal recovery in terms of the natural Signal to Interference plus Noise Ratio (SINR) criterion. There have been several partial solutions proposed in the ICA literature. It turns out that any solution to the mixing matrix reconstruction problem can be used to construct an SINRoptimal ICA demixing, despite the fact that SINR itself cannot be computed from data. That allows us to obtain a practical and provably SINRoptimal recovery method for ICA with arbitrary Gaussian noise.

Polynomial Learning of Distribution Families [link]
M. Belkin, K. Sinha, Siam Journal on Computing (SICOMP),44(4), 889911, 2015. (Short version FOCS 2010).

+ abstract

The question of polynomial learnability of probability distributions, particularly Gaussian mixture distributions, has recently received significant attention in theoretical computer science and machine learning. However, despite major progress, the general question of polynomial learnability of Gaussian mixture distributions still remained open. The current work resolves the question of polynomial learnability for Gaussian mixtures in high dimension with an arbitrary fixed number of components. The result on learning Gaussian mixtures relies on an analysis of distributions belonging to what we call "polynomial families" in low dimension. These families are characterized by their moments being polynomial in parameters and include almost all common probability distributions as well as their mixtures and products. Using tools from real algebraic geometry, we show that parameters of any distribution belonging to such a family can be learned in polynomial time and using a polynomial number of sample points. The result on learning polynomial families is quite general and is of independent interest. To estimate parameters of a Gaussian mixture distribution in high dimensions, we provide a deterministic algorithm for dimensionality reduction. This allows us to reduce learning a highdimensional mixture to a polynomial number of parameter estimations in low dimension. Combining this reduction with the results on polynomial families yields our result on learning arbitrary Gaussian mixtures in high dimensions.

Beyond Hartigan Consistency: Merge Distortion Metric for
Hierarchical Clustering [link]
Justin Eldridge, Mikhail Belkin, Yusu Wang, COLT 2015, Mark Fulk award (best student paper)!

+ abstract

Hierarchical clustering is a popular method for analyzing data which associates a tree to a
dataset. Hartigan consistency has been used extensively as a framework to analyze such clustering
algorithms from a statistical point of view. Still, as we show in the paper, a tree which is Hartigan
consistent with a given density can look very different than the correct limit tree. Specifically, Hartigan
consistency permits two types of undesirable configurations which we term oversegmentation
and improper nesting. Moreover, Hartigan consistency is a limit property and does not directly
quantify difference between trees.
In this paper we identify two limit properties, separation and minimality, which address both
oversegmentation and improper nesting and together imply (but are not implied by) Hartigan consistency.
We proceed to introduce a merge distortion metric between hierarchical clusterings and
show that convergence in our distance implies both separation and minimality. We also prove that
uniform separation and minimality imply convergence in the merge distortion metric. Furthermore,
we show that our merge distortion metric is stable under perturbations of the density.
Finally, we demonstrate applicability of these concepts by proving convergence results for two
clustering algorithms. First, we show convergence (and hence separation and minimality) of the
recent robust single linkage algorithm of Chaudhuri and Dasgupta (2010). Second, we provide
convergence results on manifolds for topological split tree clustering.

CrowdML: A PrivacyPreserving Learning Framework for a Crowd of Smart Devices
J. Hamm, A. Champion, G. Chen, M. Belkin, and D. Xuan, ICDCS 2015

+ abstract

Smart devices with builtin sensors, computational
capabilities, and network connectivity have become increasingly
pervasive. Crowds of smart devices offer opportunities to collec
tively sense and perform computing tasks at an unprecedented
scale. This paper presents CrowdML, a privacypreserving
machine learning framework for a crowd of smart devices, which
can solve a wide range of learning problems for crowdsensing
data with differential privacy guarantees. CrowdML endows
a crowdsensing system with the ability to learn classifiers or
predictors online from crowdsensing data privately with minimal
computational overhead on devices and servers, suitable for
practical largescale use of the framework. We analyze the
performance and scalability of CrowdML and implement the
system with offtheshelf smartphones as a proof of concept.
We demonstrate the advantages of CrowdML with real and
simulated experiments under various conditions.

Learning with Fredholm Kernels [link]
Qichao Que, Mikhail Belkin, Yusu Wang, NIPS 2014

+ abstract

In this paper we propose a framework for supervised and semisupervised learning
based on reformulating the learning problem as a regularized Fredholm integral
equation. Our approach fits naturally into the kernel framework and can be in
terpreted as constructing new datadependent kernels, which we call Fredholm
kernels. We proceed to discuss the “noise assumption” for semisupervised learn
ing and provide both theoretical and experimental evidence that Fredholm kernels
can effectively utilize unlabeled data under the noise assumption. We demonstrate
that methods based on Fredholm learning show very competitive performance in
the standard semisupervised learning setting.

The More, the Merrier: the Blessing of Dimensionality for Learning Large Gaussian Mixtures.
[arxiv]
Joseph Anderson, Mikhail Belkin, Navin Goyal, Luis Rademacher, James Voss, COLT 2014

+ abstract

In this paper we show that very large mixtures of Gaussians with known and identical covariance matrix are efficiently learnable in high dimension. More precisely, we prove that a mixture whose number of components is a polynomial of any fixed degree in the dimension n is polynomially learnable as long as a certain nondegeneracy condition on the means is satisfied. It turns out that this condition is generic in the sense of smoothed complexity, as soon as the dimensionality of the space is high enough. Moreover, we prove that no such condition can exist in low dimension. Our main result on mixture recovery relies on a new "Poissonization"based technique, which transforms a mixture of Gaussian to a projection of a product distribution. The problem of learning the projection can be efficiently solved using some recent results on tensor decompositions, and this gives an efficient algorithm for learning the mixture.
While our results require fixed known covariance matrix, we believe that this work is among the first steps toward better understanding the rare phenomenon of the "blessing of dimensionality" in the computational aspects of statistical inference.

Fast Algorithms for Gaussian Noise Invariant Independent Component Analysis.
[NIPS archive, GIICA code]
James Voss, Luis Rademacher, Mikhail Belkin, NIPS 2013

+ abstract

The performance of standard algorithms for Independent Component Analysis quickly deteriorates under the addition of Gaussian noise.
This is partially due to a common first step that typically consists of whitening, i.e., applying Principal Component Analysis (PCA) and
rescaling the components to have identity covariance, which is not invariant under Gaussian noise. In our paper we develop the first practical
algorithm for Independent Component Analysis that is provably invariant under Gaussian noise. The two main contributions of this work are as follows:
1. We develop and implement a more efficient version of a Gaussian noise invariant decorrelation (quasiorthogonalization) algorithm using Hessians of the cumulant functions. 2. We propose a very simple and efficient fixedpoint GIICA (Gradient Iteration ICA) algorithm, which is compatible with quasiorthogonalization, as well as with the usual PCAbased whitening in the noiseless case. The algorithm is based on a special form of gradient iteration (different from gradient descent). We provide an analysis of our algorithm demonstrating fast convergence following from the basic properties of cumulants. We also present a number of experimental comparisons with the existing methods,
showing superior results on noisy data and very competitive performance in the noiseless case.

Inverse Density as an Inverse Problem: The Fredholm Equation Approach
[arxiv]
Qichao Que, Mikhail Belkin, NIPS 2013

+ abstract

In this paper we address the problem of estimating the ratio $\frac{q}{p}$ where $p$ is a density function and $q$ is another density, or, more generally an arbitrary function. Knowing or approximating this ratio is needed in various problems of inference and integration, in particular, when one needs to average a function with respect to one probability distribution, given a sample from another. It is often referred as {\it importance sampling} in statistical inference and is also closely related to the problem of {\it covariate shift} in transfer learning as well as to various MCMC methods. It may also be useful for separating the underlying geometry of a space, say a manifold, from the density function defined on it.
Our approach is based on reformulating the problem of estimating $\frac{q}{p}$ as an inverse problem in terms of an integral operator corresponding to a kernel, and thus reducing it to an integral equation, known as the Fredholm problem of the first kind. This formulation, combined with the techniques of regularization and kernel methods, leads to a principled kernelbased framework for constructing algorithms and for analyzing them theoretically.
The resulting family of algorithms (FIRE, for Fredholm Inverse Regularized Estimator) is flexible, simple and easy to implement.
We provide detailed theoretical analysis including concentration bounds and convergence rates for the Gaussian kernel in the case of densities defined on $\R^d$, compact domains in $\R^d$ and smooth $d$dimensional submanifolds of the Euclidean space.
We also show experimental results including applications to classification and semisupervised learning within the covariate shift framework and demonstrate some encouraging experimental comparisons. We also show how the parameters of our algorithms can be chosen in a completely unsupervised manner.

Blind Signal Separation in the Presence of Gaussian Noise
[arxiv]
Mikhail Belkin, Luis Rademacher, James Voss, COLT 2013

+ abstract

A prototypical blind signal separation problem is the socalled cocktail party problem, with n people talking simultaneously and n different microphones within a room. The goal is to recover each speech signal from the microphone inputs. Mathematically this can be modeled by assuming that we are given samples from a ndimensional random variable X=AS, where S is a vector whose coordinates are independent random variables corresponding to each speaker. The objective is to recover the matrix A^{1} given random samples from X. A range of techniques collectively known as Independent Component Analysis (ICA) have been proposed to address this problem in the signal processing and machine learning literature. Many of these techniques are based on using the kurtosis or other cumulants to recover the components.
In this paper we propose a new algorithm for solving the blind signal separation problem in the presence of additive Gaussian noise, when we are given samples from X=AS + \eta, where {\eta} is drawn from an unknown ndimensional Gaussian distribution. Our approach is based on a method for decorrelating a sample with additive Gaussian noise under the assumption that the underlying distribution is a linear transformation of a distribution with independent components. Our decorrelation routine is based on the properties of cumulant tensors and can be combined with any standard cumulantbased method for ICA to get an algorithm that is provably robust in the presence of Gaussian noise. We derive polynomial bounds for sample complexity and error propagation of our method.
Our results generalize the recent work of Arora et al. which deals with a special case of ICA when S is the uniform probability distribution over the binary cube.

Graph Laplacians on Singular Manifolds: Toward understanding complex spaces: graph Laplacians on manifolds with singularities and boundaries [arxiv]
Mikhail Belkin, Qichao Que, Yusu Wang, Xueyuan Zhou, COLT 2012.

+ abstract

Recently, much of the existing work in manifold learning has been done under the assumption that the data is sampled from a manifold without boundaries and singularities or that the functions of interest are evaluated away from such points. At the same time, it can be argued that singularities and boundaries are an important aspect of the geometry of realistic data.
In this paper we consider the behavior of graph Laplacians at points at or near boundaries and two main types of other singularities: intersections, where different manifolds come together and sharp "edges", where a manifold sharply changes direction. We show that the behavior of graph Laplacian near these singularities is quite different from that in the interior of the manifolds. In fact, a phenomenon somewhat reminiscent of the Gibbs effect in the analysis of Fourier series, can be observed in the behavior of graph Laplacian near such points. Unlike in the interior of the domain, where graph Laplacian converges to the LaplaceBeltrami operator, near singularities graph Laplacian tends to a firstorder differential operator, which exhibits different scaling behavior as a function of the kernel width. One important implication is that while points near the singularities occupy only a small part of the total volume, the difference in scaling results in a disproportionately large contribution to the total behavior. Another significant finding is that while the scaling behavior of the operator is the same near different types of singularities, they are very distinct at a more refined level of analysis.
We believe that a comprehensive understanding of these structures in addition to the standard case of a smooth manifold can take us a long way toward better methods for analysis of complex nonlinear data and can lead to significant progress in algorithm design.

Data Skeletonization via Reeb Graphs
[pdf]
X. Ge, I. Safa, M. Belkin, Y. Wang, NIPS 2011.

+ abstract

Recovering hidden structure from complex and noisy nonlinear data is one of the most fundamental problems in machine learning and statistical inference. While such data is often highdimensional, it is of interest to approximate it with a lowdimensional or even onedimensional space, since many important aspects of data are often intrinsically lowdimensional. Furthermore, there are many scenarios where the underlying structure is graphlike, e.g, river/road networks or various trajectories. In this paper, we develop a framework to extract, as well as to simplify, a onedimensional "skeleton" from unorganized data using the Reeb graph. Our algorithm is very simple, does not require complex optimizations and can be easily applied to unorganized highdimensional data such as point clouds or proximity graphs. It can also represent arbitrary graph structures in the data. We also give theoretical results to justify our method. We provide a number of experiments to demonstrate the effectiveness and generality of our algorithm, including comparisons to existing methods, such as principal curves. We believe that the simplicity and practicality of our algorithm will help to promote skeleton graphs as a data analysis tool for a broad range of applications.

Convergence of Laplacian Eigenmaps [pdf, bib]
M. Belkin, P. Niyogi
preprint, short version NIPS 2008.

+ abstract

Geometrically based methods for various tasks of data analysis
have attracted considerable attention over the last few years.
In many of these algorithms, a central role is played by the
eigenvectors of the graph Laplacian of a
dataderived graph. In this paper, we show that if points are
sampled uniformly at random from an unknown submanifold ${\cal M}$
of $\R^N$, then the eigenvectors of a suitably constructed
graph Laplacian converge to the eigenfunctions of the Laplace
Beltrami operator on ${\cal M}$. This basic result directly
establishes the convergence of spectral manifold learning
algorithms such as Laplacian Eigenmaps and Diffusion Maps.
It also has implications for the understanding of geometric algorithms
in data analysis, computational harmonic analysis, geometric
random graphs, and graphics.

On Learning with Integral Operators
[pdf, bib]
L. Rosasco, M.Belkin, E. De Vito, Journal of Machine Learning Research, vol.11, pp.905934, 2010.

+ abstract

A large number of learning algorithms, for example,
spectral clustering, kernel Principal Components
Analysis and many manifold methods are
based on estimating eigenvalues and eigenfunctions
of operators defined by a similarity function or a
kernel, given empirical data. Thus for the analysis
of algorithms, it is an important problem to
be able to assess the quality of such approximations.
The contribution of our paper is twofold.
First, we use a technique based on a concentration
inequality for Hilbert spaces to provide new simplified
proofs for a number of results in spectral
approximation. Second, using these methods we
provide several new results for estimating spectral
properties of the graph Laplacian operator extending
and strengthening results from [28].

Data spectroscopy: eigenspaces of convolution operators and clustering [pdf, bib]
Tao Shi, Mikhail Belkin, Bin Yu
The Annals of Statistics, vol. 37, Number 6B (2009), 39603984.

+ abstract

This paper focuses on obtaining clustering information about a
distribution from its i.i.d. samples. We develop theoretical results to
understand and use clustering information contained in the eigenvectors
of data adjacency matrices based on a radial kernel function
with a sufficiently fast tail decay. In particular, we provide population
analyses to gain insights into which eigenvectors should be used and
when the clustering information for the distribution can be recovered
from the sample. We learn that a fixed number of top eigenvectors
might at the same time contain redundant clustering information and
miss relevant clustering information. We use this insight to design
the Data Spectroscopic clustering (DaSpec) algorithm that utilizes
properly selected eigenvectors to determine the number of clusters
automatically and to group the data accordingly. Our findings extend
the intuitions underlying existing spectral techniques such as
spectral clustering and Kernel Principal Components Analysis, and
provide new understanding into their usability and modes of failure.
Simulation studies and experiments on real world data are conducted
to show the potential of our algorithm. In particular, DaSpec is found
to handle unbalanced groups and recover clusters of different shapes
better than the competing methods.
 Towards a Theoretical Foundation for LaplacianBased
Manifold Methods [pdf, bib]
M. Belkin, P. Niyogi
Journal of Computer and System Sciences, 2008.
Volume 74, Issue 8, pp. 12891308.
Special Issue on Learning Theory, invited (short version in COLT 2005).

+ abstract

In recent years manifold methods have attracted a considerable amount of atten
tion in machine learning. However most algorithms in that class may be termed
"manifoldmotivated" as they lack any explicit theoretical guarantees. In this pa
per we take a step towards closing the gap between theory and practice for a class
of Laplacianbased manifold methods. These methods utilize the graph Laplacian
associated to a data set for a variety of applications in semisupervised learning,
clustering, data representation.
We show that under certain conditions the graph Laplacian of a point cloud of
data samples converges to the LaplaceBeltrami operator on the underlying mani
fold. Theorem 3.1 contains the .rst result showing convergence of a random graph
Laplacian to the manifold Laplacian in the context of machine learning.

Discrete Laplace Operator for Meshed Surfaces
[pdf, code, bib]
M. Belkin, J. Sun, Y. Wang, 24th Annual Symposium on Computational Geometry (SOCG) 2008.

+ abstract

In recent years a considerable amount of work in graphics and geometric optimization used tools
based on the LaplaceBeltrami operator on a surface. The applications of the Laplacian include mesh
editing, surface smoothing, and shape interpolations among others. However, it has been shown [12, 23,
25] that the popular cotangent approximation schemes do not provide convergent pointwise (or even
L^2) estimates, while many applications rely on pointwise estimation. Existence of such schemes has
been an open question [12].
In this paper we propose the first algorithm for approximating the Laplace operator of a surface from
a mesh with pointwise convergence guarantees applicable to arbitrary meshed surfaces. We show that
for a sufficiently fine mesh over an arbitrary surface, our mesh Laplacian is close to the LaplaceBeltrami
operator on the surface at every point of the surface.
Moreover, the proposed algorithm is simple and easily implementable. Experimental evidence shows
that our algorithm exhibits convergence empirically and outperforms cotangentbased methods in providing
accurate approximation of the Laplace operator for various meshes.

Consistency of Spectral Clustering [pdf, bib]
U. von Luxburg, M. Belkin, O. Bousquet,
The Annals of Statistics 2008, Vol. 36, No. 2, 555586.

+ abstract

Consistency is a key property of statistical
algorithms when the data is drawn from some
underlying probability distribution. Surprisingly,
despite decades of work, little is known about
consistency of most clustering algorithms. In this
paper we investigate consistency of the popular
family of spectral clustering algorithms, which
clusters the data with the help of eigenvectors of
graph Laplacian matrices. We develop new methods to
establish that for increasing sample size, those
eigenvectors converge to the eigenvectors of certain
limit operators. As a result we can prove that one of
the two major classes of spectral clustering
(normalized clustering) converges under very general
conditions, while the other (unnormalized clustering)
is only consistent under strong additional
assumptions, which are not always satisfied in real
data. We conclude that our analysis provides strong
evidence for the superiority of normalized spectral
clustering.

Manifold Regularization:
a Geometric Framework for Learning from Labeled and
Unlabeled Examples [pdf, bib]
M. Belkin, P. Niyogi, V. Sindhwani
Journal of Machine Learning Research, 7(Nov):23992434,
2006.

+ abstract
 We propose a family of learning algorithms based on a
new form of regularization that allows us to exploit
the geometry of the marginal distribution. We focus
on a semisupervised framework that incorporates
labeled and unlabeled data in a generalpurpose
learner. Some transductive graph learning algorithms
and standard methods including support vector
machines and regularized least squares can be
obtained as special cases. We use properties of
reproducing kernel Hilbert spaces to prove new
Representer theorems that provide theoretical basis
for the algorithms. As a result (in contrast to
purely graphbased approaches) we obtain a natural
outofsample extension to novel examples and so are
able to handle both transductive and truly
semisupervised settings. We present experimental
evidence suggesting that our semisupervised
algorithms are able to use unlabeled data
effectively. Finally we have a brief discussion of
unsupervised and fully supervised learning within our
general framework.

Heat Flow and a Faster Algorithm to Compute the
Surface Area of a Convex Body [pdf, bib]
M. Belkin, H. Narayanan, P. Niyogi, FOCS 2006.

+ abstract

We draw on the observation that the amount of heat
diffusing outside of a heated body in a short period
of time is proportional to its surface area, to
design a simple algorithm for approximating the
surface area of a convex body given by a membership
oracle. Our method has a complexity of O*(n^4), where
n is the dimension, compared to O*(n^8.5) for the
previous best algorithm. We show that our complexity
cannot be improved given the current stateoftheart
in volume estimation.

Semisupervised Learning on Riemannian Manifolds
[pdf, bib]
M. Belkin, P. Niyogi
Machine Learning, 56 (invited, special Issue on
clustering), 209239, 2004 (short version in NIPS 2002).

+ abstract

We consider the general problem of utilizing both
labeled and unlabeled data to improve classification
accuracy. Under the assumption that the data lie on a
submanifold in a high dimensional space, we develop
an algorithmic framework to classify a partially
labeled data set in a principled manner. The central
idea of our approach is that classification functions
are naturally defined only on the submanifold in
question rather than the total ambient space. Using
the LaplaceBeltrami operator one produces a basis
(the Laplacian Eigenmaps) for a Hilbert space of
square integrable functions on the submanifold. To
recover such a basis, only unlabeled examples are
required. Once such a basis is obtained, training can
be performed using the labeled data set. Our
algorithm models the manifold using the adjacency
graph for the data and approximates the Laplace
Beltrami operator by the graph Laplacian. We provide
details of the algorithm, its theoretical
justification, and several practical applications for
image, speech, and text classification.

Laplacian Eigenmaps for Dimensionality Reduction and
Data Representation [pdf, bib]
M. Belkin, P. Niyogi
Neural Computation, June 2003; 15 (6):13731396 (short version in NIPS 2001).

+ abstract

One of the central problems in machine learning and
pattern recognition is to develop appropriate
representations for complex data. We consider the
problem of constructing a representation for data
lying on a lowdimensional manifold embedded in a
highdimensional space. Drawing on the correspondence
between the graph Laplacian, the Laplace Beltrami
operator on the manifold, and the connections to the
heat equation, we propose a geometrically motivated
algorithm for representing the highdimensional data.
The algorithm provides a computationally efficient
approach to non
dimensionality reduction that
has localitypreserving properties and a natural
connection to clustering. Some potential applications
and illustrative examples are discussed.
